State the domain and range of each function.
Justify your answer
a) y=1/(x+5)
b) y=√1-2x
I know how to do domain and range, but I don't remember what shapes these form. It's not parabolas, is it? Then, to justify them, I need to draw them, so that's why I need to know the shape. Does anyone know any good site that shows the shapes and their formulas. I would really appreciate this. :)
The best way to know them is to draw them.
a. y=1/(x+5) has the same shape as 1/x, except that the graph is translated towards -x axis by 5.
http://i263.photobucket.com/albums/ii157/mathmate/sally1.png
b. y=√(1-2x) should look like √x, but mirrored about x=0.5
http://i263.photobucket.com/albums/ii157/mathmate/sally2.png
I will leave it to you to do the domains and ranges.
Oh, ok, I see. Thank you.
You're welcome!
a) To determine the domain and range of the function y=1/(x+5), we need to understand the shape of the graph.
The given function is a rational function, which means it represents a graph of a hyperbola. In this case, it is a vertical hyperbola that is symmetric with respect to the line x=-5.
To find the domain, we need to identify any x-values that would result in division by zero. In this case, if we set x+5=0, we find that x=-5. Therefore, the function is undefined at x=-5. Hence, the domain of the function is all real numbers except x = -5, or (-∞, -5) U (-5, ∞).
To find the range, we need to consider the values of y. Since the function is a rational function, the denominator cannot equal zero. Therefore, the range of the function is all real numbers except y=0, or (-∞, 0) U (0, ∞).
b) To determine the domain and range of the function y=√(1-2x), we need to understand the shape of the graph.
The given function is a square root function, which means it represents a graph of a square root curve. In this case, it is a downward opening curve.
To find the domain, we need to identify any x-values that would result in taking the square root of a negative number. Since the square root of a negative number is not a real number, we must ensure that 1-2x ≥ 0. Solving this inequality, we find that x ≤ 1/2. Therefore, the domain of the function is (-∞, 1/2] or x ≤ 1/2.
To find the range, we consider the values that y can take. Since the square root of any non-negative number is a real number, we know that the range of the function is y ≥ 0, or [0, +∞).
For a visual representation of these shapes and their formulas, you can refer to online graphing calculators such as Desmos (https://www.desmos.com) or Wolfram Alpha (https://www.wolframalpha.com). These websites allow you to input the equation and generate the graph, helping you visualize the shape and understand the domain and range of the functions.